Periodic Reporting for period 1 - UniBoGas (Universal Description of the Bose Gases)
Reporting period: 2023-08-01 to 2025-07-31
Bosons are a class of particles existing in nature which include objects of different kinds but sharing a common behaviour. Major examples are helium and rubidium and some quanta carrying forces like photons for electromagnetism. These particles exhibit Bose-Einstein Condensation (BEC): when bosons go through BEC, at extremely low temperatures they suddenly begin to act as a single quantum entity, giving rise to new states of matter with peculiar properties such as superfluidity — the ability to flow without resistance. BEC has been observed experimentally but has not a complete and rigorous mathematical understanding. In particular, our theoretical understanding of the fundamental energy laws governing Bose gases remains incomplete.
A central question is: is it possible to identity universal laws to calculate the energy of bosonic systems?
The project UniBoGas tackles this problem by providing a unified mathematical description of the energy of bosons. It focuses on whether the energy of dilute Bose gases of different kinds, including different species of bosons and interactions with repulsive and attractive forces, can be described universally, that is, independently of the detailed shape of the interaction between the particles. Main core of the project is to show, indeed, how different kinds of bosonic systems have a common pattern and behave energetically as systems of hard spheres colliding (like billiards), and the energy can be found by formulas depending only on the diameter of such spheres.
The project addresses this problem following three research lines:
• It investigates whether universal formulas can be used to calculate the energy for systems of bosons interacting through forces which are not purely repulsive. Most of the bosonic systems considered in experiments and in chemical models are included in this category.
• It shows how it is possible to calculate with universal formulas the energy of mixtures of two boson species, which are of growing experimental interest.
• It deals with the calculation of energy levels beyond the minimal energy configuration, to include excited energy states, which are crucial for linking theoretical predictions to what is measured in the lab.
Obtaining universal formulas for these broad classes of Bose gas systems greatly simplifies the evaluation of their energy, with potential impact not only in Mathematical Physics but also in fields such as Quantum Chemistry and quantum technologies. The results obtained in UniBoGas contribute to the fundamental, theoretical research on BEC, which in turn will help the technological development in quantum optics and quantum computers.
A central question is: is it possible to identity universal laws to calculate the energy of bosonic systems?
The project UniBoGas tackles this problem by providing a unified mathematical description of the energy of bosons. It focuses on whether the energy of dilute Bose gases of different kinds, including different species of bosons and interactions with repulsive and attractive forces, can be described universally, that is, independently of the detailed shape of the interaction between the particles. Main core of the project is to show, indeed, how different kinds of bosonic systems have a common pattern and behave energetically as systems of hard spheres colliding (like billiards), and the energy can be found by formulas depending only on the diameter of such spheres.
The project addresses this problem following three research lines:
• It investigates whether universal formulas can be used to calculate the energy for systems of bosons interacting through forces which are not purely repulsive. Most of the bosonic systems considered in experiments and in chemical models are included in this category.
• It shows how it is possible to calculate with universal formulas the energy of mixtures of two boson species, which are of growing experimental interest.
• It deals with the calculation of energy levels beyond the minimal energy configuration, to include excited energy states, which are crucial for linking theoretical predictions to what is measured in the lab.
Obtaining universal formulas for these broad classes of Bose gas systems greatly simplifies the evaluation of their energy, with potential impact not only in Mathematical Physics but also in fields such as Quantum Chemistry and quantum technologies. The results obtained in UniBoGas contribute to the fundamental, theoretical research on BEC, which in turn will help the technological development in quantum optics and quantum computers.
During the period of the fellowship, I managed to obtain the following main achievements corresponding to each of the three parts of the projects:
• I obtained an estimate of how strong an attraction force can be compared to the repulsion between the bosons if one wants the formula for the energy to be still analogous to the one of the hard spheres. The analysis has required to find a strategy for careful estimates from above and below for the precise energy which have given the estimates on the intensity of the attractive force.
• I proved that systems composed of two species of bosons, like compounds of Rubidium and Potassium, have an energy which can be calculated by using universal formulas which only depend on the diameters of the hard spheres reproducing inter- and intra-species collisions.
• I calculated the formula for all the low levels of energy beyond the minimal configuration for bosonic systems of hard spheres at low temperature. In a work in collaboration, we introduced new techniques to estimate the full energy by summing the contributions from localized subsystems, at the same time not losing precision on the original energy in the process.
In conclusion, the results of UniBoGas let us classify which interactions and configurations of the Bose gases admit a universal description of the energy, and obtain precise estimates of the minimal and low levels of the energy of such systems by universal formulas.
• I obtained an estimate of how strong an attraction force can be compared to the repulsion between the bosons if one wants the formula for the energy to be still analogous to the one of the hard spheres. The analysis has required to find a strategy for careful estimates from above and below for the precise energy which have given the estimates on the intensity of the attractive force.
• I proved that systems composed of two species of bosons, like compounds of Rubidium and Potassium, have an energy which can be calculated by using universal formulas which only depend on the diameters of the hard spheres reproducing inter- and intra-species collisions.
• I calculated the formula for all the low levels of energy beyond the minimal configuration for bosonic systems of hard spheres at low temperature. In a work in collaboration, we introduced new techniques to estimate the full energy by summing the contributions from localized subsystems, at the same time not losing precision on the original energy in the process.
In conclusion, the results of UniBoGas let us classify which interactions and configurations of the Bose gases admit a universal description of the energy, and obtain precise estimates of the minimal and low levels of the energy of such systems by universal formulas.
The results of UniBoGas fill some important gaps in the research on bosonic systems.
The simple universal formulas to estimate the energy for a huge variety of systems, with repulsive and attractive interactions and with multiple species of particles, widen the class of known models for which one can easily calculate the energy from few physical parameters.
The new techniques introduced, and the methods adapted to the context of the models of interest are useful tools that are now available for researchers working in the field. They can be applied to other contexts like fermion-boson mixtures and the study of formation of molecules.
The calculation of a formula for the low energy levels of the Bose gases of hard spheres can be of great importance for the analysis of experiments where excited states are usually involved.
e research on bosonic systems.
The simple universal formulas to estimate the energy for a huge variety of systems, with repulsive and attractive interactions and with multiple species of particles, widen the class of known models for which one can easily calculate the energy from few physical parameters.
The new techniques introduced, and the methods adapted to the context of the models of interest are useful tools that are now available for researchers working in the field. They can be applied to other contexts like fermion-boson mixtures and the study of formation of molecules.
The calculation of a formula for the low energy levels of the Bose gases of hard spheres can be of great importance for the analysis of experiments where excited states are usually involved.
e research on bosonic systems.